Dynamics Behaviors of a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

Figure 1

Dynamics behaviors of system (5.1) with initial conditions 𝑥(𝑛+1)=𝑥(𝑛)exp−0.8+0(0.1.0s3in5(+𝑛)0−.0(2151s.5in+(𝑛3).)5𝑦(c𝑛o)s𝑥((𝑛𝑛)))𝑥(𝑛)𝑥²(𝑛)+5.75+0.35cos√2𝑛²𝑦²(𝑛)−(0.025+0.005cos(𝑛))𝑢¹𝑦(𝑛+1)=𝑦(𝑛)exp−0.975+0.025cos√(𝑛),2𝑛+(0.195+0.005sin(√3𝑛))𝑥²(𝑛)−0.0015+0.0005cos√𝑥²(𝑛)+(5.75+0.35cos(√2𝑛))²𝑦²(𝑛)Δ𝑢₁3𝑛𝑢₂(𝑛),Δ𝑢₂(𝑛)=−(0.925+0.025sin(𝑛))𝑢₁(𝑛)+(0.0375+0.0275cos(𝑛))𝑥(𝑛),(𝑛)=−(0.925−0.025cos(𝑛))𝑢₂(𝑛)+(0.025+0.005sin(𝑛))𝑦(𝑛).(5.3) and −𝑑^𝐿+𝑓^𝑈=−0.8<0.(5.4), respectively.

(a) Species lim_𝑛→+∞𝑦(𝑛)=0.

(b) Feedback controls (𝑥(0),𝑦(0),𝑢₁(0),𝑢₂(0))^𝑇=(0.06,0.08,0.04,0.02)^𝑇